Exploring Trigonometry and the Unit Circle Module Overview

Exploring Trigonometry and the Unit Circle Module Overview

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has applications in various fields, including physics, engineering, and navigation. One fundamental concept in trigonometry is the unit circle.  In our exploration of trigonometry and the unit circle, our journey unfolds in three distinct parts.  In part one, we explore the application of trigonometric ratios within the unit circle.  We will see how these ratios aid in understanding the relationships between angles and coordinates in the circle.  In part two, we will graph sinusoidal functions and analyze the graphs. Finally, we will learn about trigonometric identities and solve equations in part three. 

Each part will be assessed independent to measure your growth and understanding of the material.

Part 1: Trigonometry and the Unit Circle

Trigonometry finds its application in the unit circle, enabling us to solve real-world problems. In this module, we will build upon your knowledge of trigonometry (sine, cosine, and tangent) and establish connections with special right triangles and the coordinate plane. This unit presents an opportunity to integrate various mathematical concepts. I hope you are thrilled to embark on this journey.

Essential Questions

• How do I think about an angle as the rotation of a ray around its endpoint?
• What is meant by the radian measure of an angle?
• What is the connection between the radian measure of an angle and the length of the arc on the unit circle the angle intercepts?

Key Terms

The following key terms will help you understand the content in this module.

Coterminal Angles  - share the same initial side and the same terminal side of angles of rotation.

Initial Side - the "beginning" side of an angle of rotation, usually on the positive x-axis

Negative Angle - an angle in standard position is negative when the location of the terminal side results from a clockwise rotation

Positive Angle - an angle in standard position is positive when the location of the terminal side results from a counterclockwise rotation

Radian - the measure of the central angle of a circle subtended by an arc of equal length to the radius

Reference Angle - the measure of the acute angle formed by the terminal side and the x-axis

Standard Position - an angle is in standard position when the vertex is at the origin and the initial side lies on the positive x-axis

Terminal Side - the "ending" side of an angle of rotation

Unit Circle - a circle with a radius of 1 and center at the origin

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Part 2: Trigonometric Expressions and Functions 

Trigonometry is a major part of Precalculus and integral in your mathematical preparation. In this module, we will connect what you know about trigonometry (sine, cosine, and tangent) to special right triangles and the coordinate plane! Precalculus is when many different parts of math finally connect! Hopefully you are excited to begin this journey.  Then we will explore graphing trigonometric functions. We will take the ratios we've learned on the Unit Circle, and use those to model the graphs of sine, cosine and tangent. Finally, you will learn how to utilize those sinusoidal functions to model real world phenomena like the tides or the temperature!

Once we explored the Unit Circle, common trigonometric ratios and how to graph the functions of sine, cosine and tangent, we will explore the inverse functions. The inverse functions can help us solve equations and use sinusoidal equations in real-world applications!

Essential Questions

• How do I think about an angle as the rotation of a ray around its endpoint?
• What is meant by the radian measure of an angle?
• What is the connection between the radian measure of an angle and the length of the arc on the unit circle the angle intercepts?
• What does it mean to prove a trigonometric identity?
• What does the Unit Circle have to do with Trigonometric Functions?
• How are the amplitude, midline, period, and phase shift of a trigonometric function related to the transformation of the parent graph?
• If I know the characteristics of the graphs of a sinusoidal function, how can I write an equation of that graph?
• How can we model a real-world situation with a trigonometric function?
• How does symmetry help us extend our knowledge of the Unit Circle to an infinite number of angles?
• How do I utilize technology to find all solutions for a trig equation?
• How do inverse trigonometric functions help us solve equations?

Key Terms

The following key terms will help you understand the content in this module.

Coterminal Angles  - share the same initial side and the same terminal side of angles of rotation.

Radian - the measure of the central angle of a circle subtended by an arc of equal length to the radius

Initial Side - the "beginning" side of an angle of rotation, usually on the positive x-axis

Terminal Side - the "ending" side of an angle of rotation

Standard Position - an angle is in standard position when the vertex is at the origin and the initial side lies on the positive x-axis

Negative Angle - an angle in standard position is negative when the location of the terminal side results from a clockwise rotation

Positive Angle - an angle in standard position is positive when the location of the terminal side results from a counterclockwise rotation

Reference Angle - the measure of the acute angle formed by the terminal side and the x-axis

Identity - an equation that is true for all values of the variable for which the expressions in the equation are defined

Unit Circle - a circle with a radius of 1 and center at the origin

Sinusoidal Function - a function is considered sinusoidal if its graph has a shape of $y=\sin x$or a transformation of $y=\sin x$.

Midline - a horizontal line located halfway between the maximum and minimum values.

Amplitude - the distance from the midline to either the maximum or minimum value; ½ the distance between the maximum and minimum values.

Period - the horizontal length of one complete cycle; the distance between any two repeating points on the function.

Frequency - the number of cycles the function completes in a given interval; the reciprocal of the period.

Asymptote - a line that continually approaches a given curve but does not meet it at any finite distance.

Phase Shift - a change in the phase of a waveform.

Even Functions - A function f is even if the graph of f is symmetric with respect to the ­­y-axis. Algebraically, f is even if and only if $f\left(-x\right)=f\left(x\right)$ for all x in the domain of f.

Odd Functions - A function f is even if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if $f\left(-x\right)=-f\left(x\right)$ for all x in the domain of f.

Inverse Function - An inverse function is a function that undoes the action of another function. A function g is the inverse of a function f if whenever y then x. In other words, applying f and then g is the same thing as doing nothing. 

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Part 3: Applying Trigonometric Identities and Equations 

In this module, we will finish our exploration of trigonometry with a deep dive into trigonometric identities! We will solve more trigonometric equations and learn new identities. Make sure you keep your Unit Circle handy!

In previous math courses, you have dealt with mostly right triangles.  However, in this module we will explore non-right, or oblique, triangles! You can still use trigonometry to solve them, but not by your traditional trigonometric ratios. Oblique triangles are often used in navigation and estimating distances - read on to see how we will solve oblique triangles!

Essential Questions

• What is an identity?
• How do I use trigonometric identities to prove statements?
• How do I use trigonometric identities to solve equations?
• How can I calculate the area of any triangle given only two sides and a non-included angle?
• How can I apply trigonometric relationships to non-right triangles?

Key Terms

The following key terms will help you understand the content in this module.

Addition Identity for Cosine - $\cos\left(x+y\right)=\cos x\cos y-\sin x\sin y$

Addition Identity for Sine - $\sin\left(x+y\right)=\sin x\cos y+\cos x\sin y$

Addition Identity for Tangent - $\tan\left(x+y\right)=\frac{\tan x+\tan y}{1-\tan x\tan y}$

Double Angle Identity for Sine - $\sin\left(2x\right)=2\sin x\cos x$

Double Angle Identity for Cosine - $\cos\left(2x\right)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$

Double Angle Identity for Tangent - $\tan\left(2x\right)=\frac{2\tan x}{1-\tan^2x}$

Half Angle Identity for Sine - $\sin\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1-\cos x}}{2}$

Half Angle Identity for Cosine - $\cos\left(\frac{x}{2}\right)=\pm\frac{\sqrt[]{1+\cos x}}{2}$

Half Angle Identity for Tangent - $\tan\left(\frac{x}{2}\right)=\pm\sqrt[]{\frac{1-\cos x}{1+\cos x}}=\frac{\sin x}{1+\cos x}$

Subtraction Identity for Cosine - $\cos\left(x-y\right)=\cos x\cos y+\sin x\sin y$

Subtraction Identity for Sine - $\sin\left(x-y\right)=\sin x\cos y-\cos x\sin y$

Subtraction Identity for Tangent - $\tan\left(x-y\right)=\frac{\tan x-\tan y}{1+\tan x\tan y}$

Even Function - a function with symmetry about the y-axis that satisfies the relationship $f\left(-x\right)=f\left(x\right)$

Odd Function - a function with symmetry about the origin that satisfies the relationship $f\left(-x\right)=-f\left(x\right)$

Pythagorean Identities -   $\cos^2\theta+\sin^2\theta=1\\ 1+\tan^2\theta=\sec^2\theta\\ 1+\cot^2\theta=\csc^2\theta\\$

Altitude of a triangle - The perpendicular distance between a vertex of a triangle and the side opposite that vertex.

Included Angle - The angle between two given sides of a triangle.

Law of Cosines - $c^2=a^2+b^2-2ab\cdot Cos(C)$

Law of Sines - $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Oblique Triangle - A triangle that is not a right triangle.

Angle of Elevation - The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

Angle of Depression - The angle below horizontal that an observer must look to see an object that is lower than the observer.

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